Goppa codes are particularly appealing for cryptographic applications.
McEliece was the first to exploit the potential of Goppa codes for the development of a secure cryptosystem [1]}.
Goppa codes are especially suited to this purpose because they have few invariants and the number of inequivalent codes grows exponentially with the length and dimension of the code.
Indeed, since the introduction of code-based cryptography by McEliece in 1978 [2]}, Goppa codes still remain among the few families of algebraic codes which resist to any structural attack. This is one of the reasons why every improvement of our knowledge of these codes is of particular interest.
In the McEliece cryptosystem, one chooses a random Goppa code as a key hence it is important that we know the number of Goppa codes for any given set of parameters. This will help in the assessment of how secure the McEliece cryptosystem is against an enumerative attack. An enumerative attack on the McEliece cryptosystem finds all Goppa codes for a given set of parameters and tests their equivalence with the public code [3]}.
There have been many attempts to count the number of Goppa codes.
